# Short Post: A New Algebra Mistake

It’s been a while since I’ve popped up for air and written anything – I’ve been a bit frazzled, plus I’ve had some self-doubt about my classes and teaching which made it difficult to find the motivation to write something. That’s all starting to fade as the semester is ending though and all our material is starting to come together and things are sinking in with my students. We’ll see though.

Anyway – I saw an interesting algebra mistake that I wanted to share. This sorta continues my curiosity with Algebra Misconceptions. We’re covering triangles and proving them congruent (those of you with knowledge of a typical geometry pace might be thinking that I’m pretty far behind where I should be… and you’d be right), but we started with the theorem that the angles in a triangle add up to 180. So I gave them this problem:

In triangle ABC,

measure of angle A = x

measure of angle B = 2x

measure of angle C = 3x

x = __________

In solving for x, most students set up the problem correctly:

x + 2x + 3x = 180

And then I kept seeing this mistake over and over from all sorts of students – even the ones with a strong algebra background. The next line looked like:

3x + 5 = 180 (Wait, what?)

It took me a long time to figure out where this came from until one of my seniors explained how he had come up with the answer. “There are 3 x’s in the problem, so I have 3x – and then there is a 2 and a 3, which together make 5. So if put them together, I get 3x’s and 2+3 = 5, so I get 3x + 5 = 180”. In other words, he was *literally* combining (counting?) the x’s in the problem, then adding the coefficients of each term.

One of the comments from my last post really got me paying closer attention to how I use language in my classroom. I think I can trace this mistake back to the phrase ‘Put all your x’s together, then put all of your constant’s together’, which I think is something I say fairly often and, if taken literally, is being done in the mistake above. Or maybe from the question ‘How many x’s do I have?’, which is another thing I ask my students. Maybe what’s missing is simply the word ‘term’ – a better question would be ‘How many x terms do I have?’, and a better phrase would be ‘combine your x terms and combine your constant terms’. Or maybe there’s even a better way to express this idea that doesn’t lead to the mistake above.

Oh – and another note: one thing I know my students still don’t quite understand is the notion that a constant next to a variable really represents multiplication. If this foundation were built, I think students would be less likely to make this kind of mistake where they separate the coefficient from the x. This is another reason I like Visual Algebra – the method of representing variables also reinforces the notion that 3x is really 3 multiplied by x, which we represent in the rectangular notion of multiplication.

So anyway… there’s some food for thought. Cheers.

Hm, I’ve never seen this one before, so I am pretty sure it is based on how you are phrasing things. I never found a way of giving the direction of combining like terms in a way I consider optimal so I’m hoping some other commenter takes up the slack.

Have the students use trial and error to solve it first so they focus on the meaning of what they are doing, mainly finding a value of x that makes the puzzle work. Then ask them to solve it procedurally (which for them is without meaning). If they make the connection, they won’t make this kind of mistake again. It’s worked like magic for me during my teaching years.

-Ihor